I need to upload 10TB of data, will this cost me?

In order to create my application I need to upload 10TB of data to an SSD, I know this would cost me 1000 bucks a month, but will this cost me any additional fees if I will upload all of this data upfront?

Bandwidth overages costs $0.02 per GB over the limit. You can verify this in FAQ - Policy.

So for 10 TB = 10,000 GB, you would pay $10,000 * $0.02 = $200.

Now, because that there is no tools available for you, to monitor your bandwidth usage, bandwidth overages are not charged.
This doesn't mean you can use it as unlimited, you should stay under your Droplet bandwidth or you risk getting locked down from your Droplet.
You must/should contact Support and see what you can do about this.

Another option is to temporary upgrade you Droplet.
Biggest, Standard Droplet - comes with 9 TB bandwidth and costs $640/mo or per hour - $0.952/hr.
Biggest, High Memory Droplet - comes with 10 TB bandwidth, but there are very expensive - $1680/mo or per hour - $2.5/hr.
Once you transfer data, you can resize to desired size you want.